part3 · mental methods unit 1 autumn first half key stage 3 national strategy ©crown copyright...
TRANSCRIPT
Square numbers
Simple sequences
Mental addition
Mental subtraction
SECTION 1
SECTION 2
SECTION 3
SECTION 4
NUMBER: PROPERTIES
MENTAL METHODS
UNIT
1 AU
TU
MN
firs
t h
alf
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PART 3
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4 hoursSUGGESTED TIME
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• Know squares to at least 10 � 10.
• Find a difference by counting up through next multiple of 10, 100, 1000.
• Add and subtract two-digit whole numbers.
• Recognise and extend number sequences formed by counting from any number
in steps of constant size, extend beyond zero when counting back.
• Generate terms of a number sequence given a rule.
• Identify mental recall and mental calculation skills that pupils need to learn and
consolidate.
• Learn square numbers to at least 10 � 10.
• Use the Star Challenges.
HOMEWORK
!
TEACHING OBJECTIVES
UNIT
1UNIT
1NUMBER: PROPERTIES
MENTAL METHODS
Square numbers
Simple sequences
Mental addition
Mental subtraction
SECTION 1
SECTION 2
SECTION 3
SECTION 4
Square numbers
You will:• learn what we mean by square numbers• recall number facts for square numbers, eg 8 � 8 = 64
� = 49
Simple sequences
You will:• recognise and extend number sequences by adding or subtracting• extend beyond zero when counting back• recognise and explain patterns in sequences
Mental addition and subtraction
You will:• add and subtract 10, 20, 30… to two-digit numbers• add and subtract 1 to 9 to any whole number• add pairs of two-digit numbers by adding tens and units separately• subtract numbers by counting up through the next multiple of
10, 100 or 1000• subtract pairs of two-digit numbers
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1UNIT
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Checklist for pupils Unit 1
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 CHECKLIST
Squares
Square numbers
PAGE
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UNIT
1
• Quickly revise odd and even numbers. You need to demonstrate the link between
square numbers and their spatial arrangements. Pupils need to have this visual picture.
• Star Challenge 2 is an activity that pupils could attempt individually and then
discuss their results in pairs.
• Learning square numbers up to 10 � 10 can be reinforced as homework.
• Square numbers provide pupils with key facts to learn multiplication tables, for
example 8 � 7 can be deduced from 7 � 7. You need to teach this and build on it
during the year.
• Discuss with pupils which square numbers they should know and when it is
appropriate to use a calculator to work out others.
• You can revisit square numbers regularly in mental sessions.
square numberKEY
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 4
1�1 2�2 3�3
SECTION 1: SQUARE NUMBERS
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1UNIT
Square numbers
Complete:
More square numbers 2
The first ten square numbers are
1, 4, . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . , . . . . . . . ,
The second square number = 4
The fourth square number = 16
The sixth square number = . . . . . . .
The tenth square number = . . . . . . .
Work out:
The thirteenth square number = . . . . . . .
The 45th square number = . . . . . . .
Complete:
The first ten square numbers1
Numbers 2 3 7 9 12 0 4 6 11
Squares 4 9 49 25 1 64 4 100
Fill in the gaps:
Squaring numbers3
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Learnsquare numbers
like 5 x 5, 7 x 7.
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Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 1
This is a table of numbers and their squares.
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Square numbers 1UNIT
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9–10 correct 1 star
19–20 correct 2 stars15–18 correct 1 star
. . . . . � . . . . . � 16
. . . . . � . . . . . � 64
. . . . . � . . . . . � 144
. . . . . � . . . . . � 169 4
3
2
1
n squared � 10
9
2 � 1 � 1
5 � 1 � 4
13 � 4 � 9
18 � 9 � 9
8 � . . . . . . � . . . . . .
10 � . . . . . . � . . . . . .
106 � . . . . . . � . . . . . .
52 � . . . . . . � . . . . . .
17 � . . . . . . � . . . . . .
50 � . . . . . . � . . . . . .
80 � . . . . . . � . . . . . .
65 � . . . . . . � . . . . . .
74 � . . . . . . � . . . . . .
104 � . . . . . . � . . . . . .
145 � . . . . . . � . . . . . .
202 � . . . . . . � . . . . . .
20 � . . . . . . � . . . . . .
25 � . . . . . . � . . . . . .
125 � . . . . . . � . . . . . .
85 � . . . . . . � . . . . . .
61 � . . . . . . � . . . . . .
169 � . . . . . . � . . . . . .
26 � . . . . . . � . . . . . .
313 � . . . . . . � . . . . . .
Squares are:
1 4 9 16
25 36 49 64 81
100 121 144 . . . . . .
. . . . . � . . . . . � 225
. . . . . � . . . . . � 484
. . . . . squared � 25
. . . . . squared � 841 8
7
6
5 . . . . . squared � 900
n squared � 7921– what is the value of n? . . . . .
10
9
STAR
CH
ALLENGE
!
Complete the square numbers:
Calculator squares search1
STAR
CH
ALLENGE
!Sums of two squares2
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 1
Write each number below as the sum of two squares.
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UNIT
1
• There are many opportunities within this section to teach and consolidate mental
addition and subtraction strategies, for example, adding and subtracting 9.
• Counting on and back in equal steps from zero helps reinforce multiples and
multiplication facts.
• Practise this work orally with written exercises used as consolidation as
appropriate.
• Demonstrate to pupils the patterns in our number system.
6 + 5 16 + 5 96 + 5 are basically the same calculation.
• Use a number line, extended below zero, as a visual prompt.
−2 −1 0 1 2 3 4
• Pupils find counting back more difficult than counting on. It needs more practice.
sequence add subtract
minus multiple
KEY
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DIRECT TEACHING POINTS
SECTION 2: SIMPLE SEQUENCES
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1UNIT
Simple sequences
Extend each sequence by counting on or back in 6s:
6 12 18 . . . . . . . . . . . . . . . . . . . .
21 27 33 . . . . . . . . . . . . . . . . . . . .
50 56 . . . . . . . . . . . . . . . . . . . . . . . . .
60 54 48 . . . . . . . . . . . . . . . . . . . .
100 94 . . . . . . . . . . . . . . . . . . . . . . . . .
52 46 . . . . . . . . . . . . . . . . . . . . . . . . .
27 21 15 . . . . . . . . . . −3 . . . . .
20 14 . . . . . . . . . . . . . . . . . . . . . . . . .8
7
6
5
4
3
2
1
Counting on and back in 6s1
Counting on and back in 9s2
Extend each sequence by counting on or back in 9s:
9 18 27 . . . . . . . . . . . . . . . . . . . .
29 38 . . . . . . . . . . . . . . . . . . . . . . . . .
63 54 . . . . . . . . . . . . . . . . . . . . . . . . .
65 56 . . . . . . . . . . . . . . . . . . . . . . . . .
27 18 . . . . . . . . . . −9 . . . . . . . . . .
30 21 . . . . . . . . . . −6 . . . . . . . . . .
50 41 . . . . . . . . . . . . . . . . . . . . . . . . .
6 �3 . . . . . . . . . . . . . . . . . . . . . . . . .8
7
6
5
4
3
2
1
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Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 2
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1UNIT
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Simple sequences
What is the rule?
Extend each sequence:
Rule: add 8 4 12 . . . . . . . . . . . . . . . . . . . . . . . . .
Rule: add . . . . . 5 12 . . . . . . . . . . . . . . . . . . . . . . . . .
Rule: add . . . . . 21 27 . . . . . . . . . . . . . . . . . . . . . . . . .
Rule: subtract 9 83 74 65 . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . 78 70 62 . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . �2 2 . . . . . . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . 18 29 . . . . . . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . 50 75 100 . . . . . . . . . . . . . . . . . . . .8
7
6
5
4
3
2
1
Rules for counting on andback3
STAR
CH
ALLENGE
!
What is the rule?
Extend each sequence:
Rule: . . . . . . . . . . . . . . . . . 43 51 . . . . . . . . . . . . . . . 83 . . . . .
Rule: . . . . . . . . . . . . . . . . . 34 45 . . . . . . . . . . . . . . . 89 . . . . .
Rule: . . . . . . . . . . . . . . . . . 150 . . . . . . . . . . 225 250 . . . . . . . . . .
Rule: . . . . . . . . . . . . . . . . . 5 30 55 . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . . . . . . 90 81 72 . . . . . . . . . . . . . . . . . . . .
Rule: . . . . . . . . . . . . . . . . . �3 3 . . . . . 15 . . . . . . . . . . 33
Rule: . . . . . . . . . . . . . . . . . �30 �21 �12 . . . . . . . . . . . . . . . 24
Rule: . . . . . . . . . . . . . . . . . 7 13 . . . . . . . . . . . . . . . . . . . . 43 8
7
6
5
4
3
2
1
50–52 correct 2 stars42–49 correct 1 star
From sequences to rules3
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 2
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1UNIT
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STAR
CH
ALLENGE
!
STAR
CH
ALLENGE
!
Simple sequences
All correct 1 star
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81
82 83 84 85 86 87 88 89 90
Look at the column with 8 at the top.
What is the rule connecting this sequence of numbers?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Start with 8.
Circle each number that is made by adding 7 onto the last number.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Describe the pattern made by the numbers you have circled.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
1
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 2
STAR
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ALLENGE
!Rules and patterns4
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UNIT
1
These sections introduce work that needs to be continually practised during the term:
addition and subtraction of two 2-digit numbers mentally.
• Pupils need to be secure in recalling addition and subtraction facts to 20.
• Counting orally forward and back over 10, 100 and 1000 boundaries can
help consolidate mental work on differences.
For example:
93 − 89
403 − 394
4003 − 3994
• A prerequisite to the introduction of written calculations is for pupils to be able
to add and subtract a pair of two-digit numbers mentally.
• You will need to teach each stage then consolidate. You can teach progression
in mental arithmetic strategies by using examples like these:
43 + 30 do this by adding 10s
56 − 20 do this by subtracting 10s
43 + 8 deduce from 13 + 8 or (43 + 7) + 143 − 8 deduce from 13 − 8 or (43 − 3) − 543 + 32 break this down into (43 + 30) + 2 or (40 + 30) + (3 + 2) no regroup
54 + 27 break this down into (54 + 20) + 7 or (50 + 20) + (4 + 7) over 10s boundary
86 + 47 break this down into (86 + 40) + 7 or (80 + 40) + (6 + 7) over 10s, 100s boundary
93 − 27 break this down into (93 − 20) − 7 across 10s boundary
64 + 39 break this down into (64 + 40) − 1 compensate
DIRECT TEACHING POINTS
89 90 93
�1 �3 �4
394 400 403
�6 �3 �9
3994 4000 4003
�6 �3 �9
SECTIONS 3 AND 4: MENTAL ADDITIONAND SUBTRACTION
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• Extend this work to cover multiples of 10 and simple decimals.
280 + 640 720 − 180 2.7 + 4.6 7.3 − 2.7
• Include examples of money £60 + £3.80 £7.20 − £3.80
and measures 5.5 m − 2.8 m
• Demonstrate the use of an empty number line as a support for pupils’ mental calculations.
63 − 38
• The visual image on a hundred square is a support for some pupils.
• Pupils need to progress so that they are not dependent on the number line or hundred square.
add addition sum
subtract minus take away difference
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�3
38 40 63
�2 �20 �25
60
43 � 25 � 68
UNIT
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Mental addition
Adding and subtracting inyour head1
23 � 8 � . . . . . . .
61 � 30 � . . . . . . .
93 � 20 � . . . . . . .
48 � 9 � . . . . . . .8
7
6
527 � 10 � . . . . . . .
68 � 7 � . . . . . . .
72 � 20 � . . . . . . .
37 � 8 � . . . . . . .4
3
2
1 48 � 20 � . . . . . . .
74 � 7 � . . . . . . .
86 � 50 � . . . . . . .
97 � 8 � . . . . . . .12
11
10
9
IDEA
74 � 28 � . . . . . . .
65 � 33 � . . . . . . .
27 � 43 � . . . . . . .
25 � 35 � . . . . . . .8
7
6
5
Example
23 � 35 � . . . . . . .
37 � 42 � . . . . . . .
42 � 34 � . . . . . . .
61 � 27 � . . . . . . .4
3
2
1 19 � 21 � . . . . . . .
55 � 37 � . . . . . . .
44 � 48 � . . . . . . .
48 � 67 � . . . . . . .12
11
10
9
Write down the answers to each of these:
Adding pairs of numbers inyour head2
Work out 45 � 36
40 + 30 = 70and 5 + 6 = 11
70 + 11 = 81
45 + 30 = 75and 75 + 6 = 81 Which way do you like best?
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 3
Write down the answers to each of these:
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1UNIT
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Subtracting across the tensboundary1
Mental subtraction
101 � 97 � . . . . .
84 � 78� . . . . .8
7 36 � 25� . . . . .
97 � 85� . . . . .10
9
54 � 49 �
83 � 76 �
75 � 68 �
46 � 34 �4
3
2
1
53 � 44� . . . . .
86 � 79� . . . . .6
5
Now do it in your head!
Example
89 90 93
�1 �3 �4Work out 93 � 89 by
counting on. Use an
empty number line. 93 � 89 � 4
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 4
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1UNIT
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Mental subtraction
Bigger jumps across thetens boundaries2
74 � 49 �
95 � 36 �
63 � 16 �
63 � 34 �
84 � 39 �
105 � 67 � 6
5
4
3
2
1
Work these out in your head.
76 � 39 � . . . . .
75 � 37 � . . . . .10
9 54 � 36 � . . . . .
71 � 47 � . . . . .12
1182 � 45 � . . . . .
83 � 55 � . . . . .8
7
�3Example
38 40 63
�2 �20 �25Work out 63 � 38 by
counting on. Use an
empty number line. 63 � 38 � 2560
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 4
Mental subtraction
Subtracting across thehundreds boundary3
311 � 295 � . . . . .
702 � 691 � . . . . .8
7 415 � 397 � . . . . .
904 � 895 � . . . . .10
9
204 � 192 �
510 � 499 �
706 � 690 �
612 � 595 �4
3
2
1
602 � 588 � . . . . .
305 � 299 � . . . . .6
5
PAGE
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1UNIT
Subtracting across thethousands boundary4
3004 � 2990 � . . . . .
5010 � 4994 � . . . . .8
7 6012 � 5980 � . . . . .
3015 � 2990 � . . . . .10
9
3004 � 2995 �
7010 � 6992 �
8006 � 7990 �
9009 � 8998 �4
3
2
1
1011 � 997 � . . . . .
2005 � 1997 � . . . . .6
5
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Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 4
1UNIT
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Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 SECTION 4
STAR
CH
ALLENGE
!
STAR
CH
ALLENGE
!
Mental subtraction
12–15 correct 2 stars8–11 correct 1 star
Bigger jumps across thehundreds boundaries6
525 � 475
721 � 570
82 � 55
73 � 378
7
6
565 � 42
312 � 280
95 � 59
72 � 234
3
2
1 540 � 460
86 � 68
650 � 580
842 � 79812
11
10
9
Work these out in your head.
Counting on mixture5
Example
283 300 700 705
�17 �400 �5 �422
705 � 283 � 422
9 correct 2 stars7–8 correct 1 star
Work out 705 � 283
by counting on. Use
an empty number line.
265 � 88
810 � 4934
3435 � 190
620 � 3702
1 535 � 275
825 � 2956
5
651 � 5928420 � 1997 922 � 6889
Now do these in your head.
Unit1Answers
The first ten square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
More square numbers
4, 16, 36, 100, 169, 2025
Squaring numbers
2 3 7 9 5 12 0 1 8 4 6 2 10 11
4 9 49 81 25 144 0 1 64 16 36 4 100 121
Section 1
3
2
1
Square numbers
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1UNIT
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Counting on and back in 6s
6 12 18 24 30 36 42
21 27 33 39 45 51 57
50 56 62 68 74 80 86
60 54 48 42 36 30 24
100 94 88 82 76 70 64
52 46 40 34 28 22 16
27 21 15 9 3 �3 �9
20 14 8 2 �4 �10 �168
7
6
5
4
3
2
1
Section 2
1
Simple sequences
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 ANSWERS
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Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 ANSWERS
Unit1Answers
Counting on and back in 9s
9 18 27 36 45 54 63
29 38 47 56 65 74 83
63 54 45 36 27 18 9
65 56 47 38 29 20 11
27 18 9 0 �9 �18 �27
30 21 12 3 �6 �15 �24
50 41 32 23 14 5 �4
6 �3 �12 �21 �30 �39 �488
7
6
5
4
3
2
1
2
Simple sequences continued
Rules for counting on and back
add 8 4 12 20 28 36 44 52
add 7 5 12 19 26 33 40 47
add 6 21 27 33 39 45 51 57
subtract 9 83 74 65 56 47 38 29
subtract 8 78 70 62 54 46 38 30
add 4 �2 2 6 10 14 18 22
add 11 18 29 40 51 62 73 84
add 25 50 75 100 125 150 175 200
Mental addition
Adding and subtracting in your head
8
7
6
5
4
3
2
1
Section 3
1
3
37
75
92
454
3
2
1 31
91
73
398
7
6
5 28
67
36
8912
11
10
9
Adding pairs of numbers in your head
Subtracting across tens boundary
Bigger jumps across the tens boundaries
Subtracting across the hundreds boundary
Subtracting across the thousands boundary
Key
Stag
e 3
Nat
iona
l Str
ateg
y
©CR
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OPY
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2001
1UNIT
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continued
Unit1Answers
58
79
76
884
3
2
1 102
98
70
608
7
6
5 40
92
92
11512
11
10
9
5
7
7
124
3
2
1 9
7
4
68
7
6
5 11
1210
9
12
11
16
174
3
2
1 14
6
16
118
7
6
5 18
910
9
25
592
1 45
386
5 37
3810
9
47
294
3 37
288
7 18
2412
11
Section 3
Section 4
4
3
2
1
2
Mental addition
Mental subtraction
9
18
16
114
3
2
1 14
8
14
168
7
6
5 32
2510
9
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 ANSWERS
Key
Stag
e 3
Nat
iona
l Str
ateg
y
©CR
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2001
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63
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 ANSWERS
Unit1Answers
2
1
Star Challenge answers
22 � 22 � 484
5 squared � 25
29 squared � 841
30 squared � 900
n � 8910
9
8
7
64 � 4 � 16
8 � 8 � 64
12 � 12 � 144
13 � 13 � 169
15 � 15 � 2255
4
3
2
1
Calculator squares searchSTAR
CH
ALLENGE
!
2 � 1 � 1
5 � 1 � 4
13 � 4 � 9
18 � 9 � 9
8 � 4 � 4
10 � 1 � 9
106 � 81 � 25
52 � 36 � 16
20 � 16 � 4
25 � 16 � 9
125 � 121 � 4
85 � 81 � 4
61 � 36 � 25
169 � 144 � 25
26 � 25 � 1
313 � 144 � 169
17 � 1 � 16
50 � 1 � 49
or 25 � 25
80 � 64 � 16
65 � 1 � 64
74 � 49 � 25
104 � 100 � 4
145 � 1 � 144
or 64 � 81
202 � 81 � 121
Sums of two squaresSTAR
CH
ALLENGE
!
9–10 correct 1 star
19–20 correct 2 stars15-18 correct 1 star
PAGE
64
Key
Stag
e 3
Nat
iona
l Str
ateg
y
©CR
OW
N C
OPY
RIG
HT
2001
1UNIT
PAGE
64
Rules and patterns
add 9
sloping lines going down to the left
Counting on mixture
23 49 27 18
32 50 36 70
36 151 80 44
Bigger jumps across the hundreds boundaries
245 317 619
250 260 1243
177 530 234963
852
741
12963
11852
10741
3
2
1
Unit1Answers
Star Challenge answers continued
From sequences to rules
add 8 43 51 59 67 75 83 91
add 11 34 45 56 67 78 89 100
add 25 150 175 200 225 250 275 300
add 25 5 30 55 80 105 130 155
subtract 9 90 81 72 63 54 45 36
add 6 �3 3 9 15 21 27 33
add 9 �30 �21 �12 �3 6 15 24
add 6 7 13 19 25 31 37 438
7
6
5
4
3
2
1
6
5
4
3STAR
CH
ALLENGE
!
STAR
CH
ALLENGE
!
STAR
CH
ALLENGE
!
STAR
CH
ALLENGE
!
Key Stage 3 National Strategy
SPRINGBOARD 7
PART 3 UNIT 1 ANSWERS
9 correct 2 stars7-8 correct 1 star
50-52 correct 2 stars42-49 correct 1 star
12-15 correct 2 stars8-11 correct 1 star
All correct 1 star